Tight --Frame and Its Novel Characterizations via Atomic Systems

1School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China

Academic Editor: Remi Léandre

Received02 Nov 2015

Revised05 Jan 2016

Accepted06 Jan 2016

Published26 Jan 2016

Abstract

-frame is a generalization of -frame. We generalize the tight -frame to --frame via atomic systems. In this paper, the definition of tight --frame is put forward; equivalent characterizations and necessary conditions of tight --frame are given. In particular, the necessary and sufficient condition for tight --frame being tight -frame is obtained. Finally, by means of methods and techniques of frame theory, several properties of tight --frame are given.

1. Introduction

Frame in Hilbert space was first introduced in 1952 by Duffin and Schaeffer [1] to deal with nonharmonic Fourier series and reintroduced in 1986 by Daubechies et al. [2]. Since then the frame theory began to be more wildly studied. Today, frame theory has been widely used in filter theory [3], image processing [4], numerical analysis, and other areas. We refer to [5–8] for an introduction to frame theory in Hilbert space and its application.

With the deepening of research on frame theory, various generalizations of frames have been proposed; see [9–12]. Atomic systems for subspaces were first introduced by Feichtinger and Werther in [13] based on examples arising in sampling theory. In 2011, Găvruţa [14] introduced -frame in Hilbert spaces to study atomic decomposition systems and discussed some properties of them. In [15–18], some conclusions of -frame were given. With the extensive research of -frame and -frame in Hilbert space, Zhu et al. [19, 20] began to study --frame, which was limited to the range of a bounded linear operator in Hilbert space and had gained greater flexibility in practical application relative to -frame. --frame, as a more general frame than -frame and -frame, has become one of the most active fields in frame theory in recent years. In [20, 21], several properties and characterizations of --frame were obtained. However, many problems of --frame have not been studied. Based on these important results of --frame, we extend tight -frame to --frame and put forward the concept of tight --frame. In this paper, we give equivalent characterizations and necessary conditions of tight --frame for Hilbert space. We also obtain the necessary and sufficient condition of tight --frame to be tight -frame. Finally, we present several properties of tight --frame for Hilbert space.

Throughout this paper, is separable Hilbert space and is the identity operator. is a collection of all bounded linear operators from to , where and are two Hilbert spaces. In particular, is a collection of all bounded linear operators from to . For any , is the range of and is the adjoint operator of . is a sequence of closed subspaces of , where is a subset of integers . is defined by with the inner product given by It is clear that is a complex Hilbert space.

2. Preliminaries

In this section, some necessary definitions and lemmas are introduced.

Definition 1 (see [9, Definition 1.1]). A sequence is called a -frame for with respect to if there exist two positive constants and such that, for all ,The constants and are called the lower and upper bounds of -frame, respectively. If the right inequality is satisfied, then is said to be a -Bessel sequence for with respect to . If , we call this -frame a tight -frame, and if , it is called a Parseval -frame.For a -Bessel sequence , defines a bounded linear operator, that is, The adjoint operator is given by By composing with its adjoint , we obtain the bounded linear operator We call , , and the preframe operator, analysis operator, and frame operator of -Bessel sequence, respectively.

Definition 2 (see [22, Definition 2.6]). We say is -orthonormal basis for with respect to , if it is -biorthonormal with itself, that is, , , , , and for any one has .

Definition 3 (see [21, Theorem 2.5]). Let and for any . A sequence is called a --frame for with respect to if there exist constants such that The constants and are called the lower and upper bounds of --frame, respectively.

Remark 4. Every --frame is a -Bessel sequence for with respect to . If , then --frame is just the ordinary -frame.

Motivated by the definition of tight -frame, we give the following definition of tight --frame.

Definition 5. Let and for any . A sequence is called a tight --frame for with respect to if there exists constant such that The constant is called the bound of tight --frame. If , we call this tight --frame a Parseval --frame for with respect to .

Definition 7. Let . An operator is said to be left-invertible if there exists an operator such that . The operator is called a left-inverse of ; that is, . Similarly, an operator is said to be right-invertible if there exists an operator such that . The operator is called a right-inverse of ; that is, . If there exists such that , then we say that is invertible and is an inverse of ; that is, .

Lemma 8 (see [23, Theorem 1]). Let and . The following conditions are equivalent:(1).(2)There exists such that .(3)There exists a bounded operator so that .

Lemma 9 (see [21, Theorem 2.5]). Let . Then the following statements are equivalent:(1) is a --frame for with respect to .(2) is a -Bessel sequence for with respect to and there exists a -Bessel sequence for with respect to such that

3. Properties of Tight --Frame for Hilbert Space

In this section, we first give characterizations of tight --frame and then give several properties of tight --frame.

Theorem 10. Let ; , and let be the preframe operator of . Then the following statements are equivalent:(1) is a tight --frame for with respect to with bound .(2)There exists constant such that for any .(3)There exists constant such that .

Proof. . Suppose that is a tight --frame for with respect to with bound . By the definition of tight --frame, we get Since is the preframe operator of , we have This implies that for any .. If there exists constant such that for any , then we obtain Hence .. If there exists constant such that , then , . That is, Therefore, is a tight --frame for with respect to with bound . The proof of Theorem 10 is completed.

Corollary 11. Suppose that is a tight --frame for with respect to with bound , is the preframe operator of , and is the frame operator of ; then(1);(2);(3).

Proof. Theorem 10 together with Lemma 8 shows that (1) and (2) are satisfied. We only need to prove that (3) holds. Assume that is a tight --frame for with respect to with bound . By Theorem 10, we have , . Therefore,The proof of Corollary 11 is completed.

Lemma 9 gives an equivalent characterization of --frame; does the tight --frame have the similar characterization? Clearly, if is a tight --frame for with respect to , then there exists a -Bessel sequence for with respect to such that for any . The theorem below gives a necessary condition of tight --frame.

Theorem 12. Let . Suppose that is a tight --frame for with respect to with bound . Then there exists a -Bessel sequence for with respect to with bound such that , , and .

Proof. Let be a tight --frame for with respect to . By Lemma 9, there exists a -Bessel sequence for with respect to such that , . For any , we have Via (15), Since is a tight --frame for with respect to with bound , we get Furthermore, Therefore, . The proof of Theorem 12 is completed.

Note that when , tight --frame is tight -frame. One may wonder whether when tight --frame is tight -frame as well. In fact, the answer is negative. The following example demonstrates this.

Example 13. Suppose that ; . Let be an orthonormal basis of , and let . Now define and as follows:By a simple calculation, we haveFor any , we have Obviously, and are Parseval -frames for with respect to .Define the bounded linear operator as follows: Now we prove that is a tight --frame for with respect to . For any , we have Hence, . It follows that Therefore, for any , we have . Via the definition of tight --frame, is a tight --frame for with respect to . For any , we have .Example 13 shows that if a tight --frame is a tight -frame, then cannot be . In the following theorem, we state a necessary and sufficient condition for a tight --frame being a tight -frame.

Theorem 14. Let and . Suppose that is a tight --frame for with respect to with bound . Then is a tight -frame for with respect to with bound if and only if is right-invertible and the right-invertible operator is .

Proof. First, we prove the sufficient condition. Since is a tight --frame for with respect to with bound , we have Assume that is a tight -frame for with respect to with bound . Then, for any , we get By (25) and (26), for any , implying that . Then, for any , we have . This implies that . So is right-invertible and the right-invertible operator is .Next, we prove the necessary condition. Suppose that is right-invertible and the right-invertible operator is . Then ; that is, . So That is, Since is a tight --frame for with respect to with bound , we have This implies that is a tight -frame for with respect to with bound . The proof of Theorem 14 is completed.

In the following, we will verify whether the in Example 13 is equal to . For any , we have It follows that . This implies that is right-invertible and the right-invertible operator is .

Corollary 15. Let . Suppose that is a Parseval --frame for with respect to ; then is a Parseval -frame for with respect to if and only if is right-invertible and the right-invertible operator is .

To enrich the theory of atomic systems, we give several important properties of tight --frame in the following.

Theorem 16. Let . If is a tight -frame for with respect to with bound , then is a tight --frame for with respect to with bound .

Proof. Since is a tight -frame for with respect to with bound , we have Again, for any , we have ; then Therefore, is a tight --frame for with respect to with bound . The proof of Theorem 16 is completed.

Corollary 17. Let . If is a -orthonormal basis for with respect to , then is a tight --frame for with respect to .

Theorem 18. Let . If is a tight --frame for with respect to with bound , then is a tight --frame for with respect to with bound .

Proof. Since is a tight --frame for with respect to with bound , we have And, for any , we have ; then So is a tight --frame for with respect to with bound . The proof of Theorem 18 is completed.

Corollary 19. Let . If is a tight --frame for with respect to , then is a tight --frame for with respect to , where is a given positive integer.

Theorem 20. Let , and let be the collection of all tight --frames for with respect to . Then if and only if there exists such that .

Proof. . If is a -orthonormal basis for with respect to , by Corollary 17, we get that is a tight --frame for with respect to . Since , we have that is a tight --frame for with respect to . Assume that the bound of tight --frame is . By the definition of tight --frame, we obtain Since is a -orthonormal basis for with respect to , we have for any . By the definition of -orthonormal basis, we get By (35) and (36), we get for any . So .. Suppose that is a tight --frame for with respect to with bound ; then Under this assumption, there exists such that . So for any , we have . Hence Therefore, is a tight --frame for with respect to with bound . The proof of Theorem 20 is completed.

Corollary 21. Let and let be the collection of all Parseval --frames for with respect to . Then if and only if .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grants nos. 61261043 and 10961001) and Natural Science Foundation of Ningxia (Grant no. NZ13084).

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